Weak Coupling Theory

Introduction

The theory of weak coupling (Kuramoto 1984; Kopell and Ermentrout 2002; Brown et al. 2004; Izhikevich 2007; Schwemmer and Lewis 2012) has been extremely helpful in developing our understanding of synchronization in neuronal networks. The theory allows one to reduce the dynamics of a weakly coupled network of neuronal oscillators to a system of equations involving only the phase of each oscillator. The resulting “phase models” are given in terms of the neurons’ infinitesimal phase response curves (iPRCs) and the input to the neuron via coupling. This article provides an explanation of how the phase models are derived and gives an example of how the theory can be used to determine phase-locking in a pair of neurons connected by reciprocal inhibition.

Model Neuronal Networks and Phase Models

A network of M weakly coupled heterogeneous neurons can be represented by

$$ \frac{d{X}_j}{ dt}=F\left({X}_j\right)+\varepsilon {\tilde{F}}_j\left({X}_j\right)+{\displaystyle...

Appendix 1: Computing the iPRC Using the Adjoint Method

As mentioned previously, the major practical asset of the singular perturbation approach to deriving phase models (Malkin 1956; Neu 1979; Ermentrout 1981) is that it provides a method to efficiently compute the iPRC for model neurons. Specifically, the iPRC is a T-period solution to

$$ \frac{ dZ}{ dt}=- DF{\left({X}_{LC}(t)\right)}^TZ $$

(16)

subject to the normalization constraint

$$ Z(0)\cdot {X}_{LC}^{\prime }(0)=1. $$

(17)

This equation is the adjoint equation for the unperturbed model neuron (Eq. 1) with ε = 0, i.e., $\frac{ dX}{ dt}=F(X)\Big)$ linearized around the limit cycle solution X _LC (t).

In practice, the solution to Eq. 16 is found by integrating the equation backwards in time (Williams and Bowtell 1997). The adjoint system has the opposite stability of the original system Eq. 1, which has an asymptotically stable T-periodic limit cycle solution. Thus, we integrate backwards in time from an arbitrary initial condition so as to dampen out the transients and arrive at the (unstable) periodic solution of Eq. 16. To obtain the iPRC, we normalize the periodic solution using Eq. 17. This algorithm is automated in the software package XPPAUT (Ermentrout 2002), which is available for free on Bard Ermentrout’s webpage www.math.pitt.edu/~bard/bardware/.